A food processing company is developing a new product with milk as a key ingredient. As a part of the process, milk is heated by water in a countercurrent double pipe heat exchanger. The temperature needs to be accurately regulated to provide the downstream unit operations with milk at the expected temperature and consistency.

Rather than running time-consuming and expensive experiments, the engineer decided to model the heat exchanger and the control loop using software. This process, however, demands an accurate model that takes into account the temperature dynamics across the entire length of the countercurrent heat exchanger, so that the controller can be reliably tuned.
A computational fluid dynamics approach was considered, but rejected because model creation would be time consuming and costly, and the results difficult to integrate into a control loop. Ideally, what was required was a single environment in which a heat exchanger could be rapidly prototyped, and different control strategies easily explored.

The engineer found MapleSim to be the ideal environment to model both the plant and controller. A physical model of the heat exchanger was developed rapidly. The temperature dynamics were modeled by dividing the heat exchanger into a given number of control volumes.
A heat balance on a typical control volume resulted in three differential equations - one each for the tube- and shell-side liquids, and one to model the heat capacity of the tube wall. Axial heat flow along the tube wall was also modeled.

As the temperature changes, the viscosity of the tube-side milk changes significantly along the length of the exchanger. Since the heat transfer coefficient (calculated by the Dittus-Boelter correlation) varies with the viscosity, an empirical correlation describing the milk viscosity as a function of temperature was hence included in the system equations.

Using natural math notation, all the governing equations for a single control volume were entered into a custom component template in MapleSim. The entire set of equations for all control volumes were then programmatically generated using Maple’s symbolic programming language, as part of the component’s definition . This approach meant a finer discretization (by increasing the number of control volumes) could be explored by changing a single parameter, without risking the introduction of human error into potentially hundreds of equations. Additionally, because the equations governing the component were in natural math notation, the custom component was highly auditable.

The control system could now be prototyped, with the temperature of the tube-side milk governed by regulating the flow rate of the shell-side hot water stream. Thermocouples, modeled by fitting the temperature-voltage data from the manufacturer to a linear equation using a custom component, monitor the temperatures of the outlet streams. These output a voltage to a controller. The controller in turn outputs a control voltage, which determines the open position of a valve. A pressure reading across the valve then determines the flow of the shell-side water stream.

With this high fidelity model, the engineer was able to tune the controller with great accuracy, ensuring the milk is brought to the required temperature even when upstream disturbances cause variations in temperatures and flow rates as the liquids enter the heat exchanger. Working in a single environment meant the work was completed quickly and inexpensively. In addition, since the model is acausal, it can be easily reconfigured to control other aspects of the heat exchanger, such as regulating the temperature of the shell-side stream instead of the tube-side stream. Since the custom component interface automates the process of generating the governing equations for the entire heat exchanger, more complex physical effects, such as heat loss to the environment, can be easily incorporated simply by adjusting the heat balance equations for a single control volume. Comprehensive documentation describing the derivation of the system equations and their physical principles was included as part of the model, so other engineers will be able to quickly review, extend, and enhance the model, even if the original designer is unavailable.

The MapleSim model of the heat exchanger, with custom components for the heat exchanger, valve, and thermocouples.

Typical results. The tubeside flowrate starts at 0.001 m3s-1, but increases to 0.002 m3s-1 at 600s. The shell-side flow is regulated to maintain the tube-side outlet temperature at the appropriate value.

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